Inverse Scattering Methods and the1NExpansion Applied to Field Theory in More than Two Dimensions

Space-dependent extrema of the one-loop effective action in quantum field theory are found with use of scattering variables in the angular momentum. The $N$-component $g{\left({\overrightarrow{\varphi }}^2\right)}^2$ theory in three space dimensions is studied in detail. These instantons are classified in "homotopy classes" by "Regge poles." A comprehensive picture of them is obtained from exact results in the limits $g\to 0^{+}$ and $g\to \infty$ and from variational calculations. The large-order behavior of the $\frac{1}{N}$ series is obtained.